Hyperspectral Unmixing via L1/2 Sparsity-Constrained Nonnegative Matrix Factorization

Hyperspectral unmixing is a crucial preprocessing step for material classification and recognition. In the last decade, nonnegative matrix factorization (NMF) and its extensions have been intensively studied to unmix hyperspectral imagery and recover the material end-members. As an important constraint for NMF, sparsity has been modeled making use of the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">L₁</tex></formula> regularizer. Unfortunately, the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">L₁</tex></formula> regularizer cannot enforce further sparsity when the full additivity constraint of material abundances is used, hence limiting the practical efficacy of NMF methods in hyperspectral unmixing. In this paper, we extend the NMF method by incorporating the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">L1/2</tex></formula> sparsity constraint, which we name <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">L1/2</tex> </formula> -NMF. The <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">L1/2</tex></formula> regularizer not only induces sparsity but is also a better choice among <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">Lq(0 < q < 1)</tex> </formula> regularizers. We propose an iterative estimation algorithm for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX"> L1/2</tex></formula> -NMF, which provides sparser and more accurate results than those delivered using the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">L₁</tex></formula> norm. We illustrate the utility of our method on synthetic and real hyperspectral data and compare our results to those yielded by other state-of-the-art methods.